Page 41 - MetalForming November 2022
P. 41

 Cutting Edge
By Eren Billur, Ph.D.
Preparing the Hardening Curve
Ahardening curve, also known as the true plastic strain vs. true stress curve and essentially only a tensile-test result tweaked with some formulas, is required for all metal form- ing simulations. Most commercially available software would ask the user to enter this data until reaching 1.0 true plastic strain ( ɛp) or 178-percent engineering plastic strain. No automo- tive sheet material can elongate that much at room temperature, so what should we do?
the Hardening Curve
A tensile test involves recording force and extension values (Fig. 1a), with this data then converted to an engineering stress-strain curve by dividing the force by the initial cross- section area and the extension by the original gauge length. Most of the material specifications (elastic modu- lus, yield stress, universal tensile strength, uniform and total elongation) are defined on the curve as shown in Fig. 1b (see recent Metal Matters columns in MetalForming for detailed explanations of these specifications).
Eren Billur is the founder of Billur Metal Form, a consulting, engineering and training company in Ankara, Turkey. He stud- ied at Baskent University and Virginia Common- wealth University, received a Ph.D. in Mechanical Engineering from The Ohio State Uni-
versity, and worked as a researcher at the Center for Precision Forming. His areas of expertise include material characterization, sheet metal forming processes and finite element simulations. He has authored/co-authored more than 20 scien- tific papers (including proceedings) and con- tributed to four books, including “Hot Stamping of Ultra High Strength Steels,” published in 2018. Eren Billur
Billur Metal Form, Founder
Engineering stress-strain curves gain wide use for design purposes. However, analysis of plastic deforma- tion, such as with metal forming processes, requires true stress-strain curves. True stress-true strain curves can be drawn only until reaching the end of uniform elongation (shown as a square in Fig. 1c). The data gathered after necking must be deleted (dashed line), as the formula used to develop the curve does not remain valid in this region. Lastly, we take the elastic strain out to reveal the true plastic strain- true stress curve, or hardening curve (Fig. 1d).
For a typical automotive sheet metal, the hardening curve (Fig. 1d) can be determined only until 0.15 true plastic strain. For deep-draw-quality steels, the maximum true plastic strain can slightly exceed 0.25. Most software, however, requires this data until 1.0 true plastic strain. The most common method to achieve this: Extrapolate the data by fitting an equation to the data you know, and trying to predict
how the true stress may evolve at high (true plastic) strains.
Material Models
Two main classes of material models exist: unbounded and saturation mod- els. Generally, we use unbounded mod- els for most steels and saturation mod- els for aluminum. Unbounded models, unlike saturation models, feature a strain-hardening exponent “n” (Table 1). You may have heard that n is not con- stant for aluminum, or that aluminum cannot have a single n-value for alu- minum. Also, advanced high-strength steel (AHSS) grades do not have a con- stant n-value.
More recently, mixed models have gained use for steels and aluminum alloys. These mixed models typically contain a ratio factor (shown by α or μ). Commonly used mixed models include Hollomon/Voce and Swift/Hockett- Sherby, with the former as the default hardening-curve model for the Yoshi- da-Uemori material model used in advanced springback prediction.
How to decide on a model? Use unbounded for low-strength steels, satura- tion for aluminum and mixed models for AHSS, or check all models to see which fits best. In any case, different models may yield huge variations at true plas- tic strain of 1. Fig. 2 shows almost a 20-percent differ- ence between two material models. With no further
 Table 1—Common Material Models
Hollomon or Ludwik
σ = Kɛn
 σ = K(ɛp + ɛ0)n
  σ = K(ɛp + ɛ0)n - D
 σ = σsat-(σsat-σyield) e-
 Hockett- Sherby
σ = σsat-(σsat-σyield) e-
     Table 2—Common Mixed Material Models
 Hollomon/Voce (also known as Yoshida-Uemori)
σ = α[Kɛn] + (1-α)[σsat-(σsat-σyield) e- ]
 σ =α[K(ɛp+ɛ0)n] + (1-α)[σsat-(σsat-σyield) e- ]
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